A 16.0-N weight slides down a rough inclined plane that makes an angle of 28.0° with the horizontal. The weight, which started from rest, gains a speed of 15.0 m/s after sliding 150 m. How much work is done against friction?
use the work-energy theorem
determine mass = 16.0N/9.8 m/s^2
initial velocity = 0; so initial KE = 0
final velocity = 15.0 m/s
final height = 0; so final PE = 0
initial height = 150 sin 28.0 degrees
substitute values in the work-energy theorem expression
To calculate the work done against friction, we need to find the net work done on the weight (W_net) and subtract the work done by gravity (W_gravity).
1. Net Work (W_net):
The net work done on an object is equal to the change in its kinetic energy. We can calculate the net work using the equation:
W_net = (1/2) * m * (v^2 - u^2)
where m is the mass of the weight, v is the final velocity, and u is the initial velocity.
Given:
Mass of the weight (m) = 16.0 N (weight is a force, not mass, but we assume the mass is 16.0 kg as weight = mass * acceleration due to gravity, where acceleration due to gravity is approximately 9.8 m/s^2)
Final velocity (v) = 15.0 m/s
Initial velocity (u) = 0 (starting from rest)
Substituting the values into the equation:
W_net = (1/2) * 16.0 kg * (15.0 m/s)^2
W_net = 2.0 kg * (225 m^2/s^2)
W_net = 450.0 J
2. Work done by gravity (W_gravity):
The work done by gravity is given by the equation:
W_gravity = m * g * d * cos(θ)
where g is the acceleration due to gravity and d is the displacement along the inclined plane.
Given:
Acceleration due to gravity (g) ≈ 9.8 m/s^2
Displacement along the inclined plane (d) = 150 m
Angle of the inclined plane (θ) = 28.0°
Substituting the values into the equation:
W_gravity = 16.0 kg * 9.8 m/s^2 * 150 m * cos(28.0°)
(Notice that the weight in this case is equal to the mass multiplied by the acceleration due to gravity)
W_gravity = 15680.0 N * m * cos(28.0°)
W_gravity ≈ 13686.8 J
3. Work done against friction:
To find the work done against friction, we subtract W_gravity from W_net:
Work against friction = W_net - W_gravity
Work against friction = 450.0 J - 13686.8 J
Work against friction ≈ -13236.8 J (negative sign indicates work done against the direction of motion, i.e., friction)
Therefore, approximately 13236.8 J of work is done against friction.
To find the work done against friction, we need to determine the net work done on the object and subtract the work done by other forces. The net work done is equal to the change in kinetic energy.
First, let's calculate the change in kinetic energy:
The initial speed of the weight, u = 0 m/s (as it starts from rest).
The final speed of the weight, v = 15.0 m/s.
The mass of the weight, m = weight/g, where g is the acceleration due to gravity (approximately 9.8 m/s^2).
The weight is given as 16.0 N, so m = 16.0 N / 9.8 m/s^2 = 1.63 kg.
The change in kinetic energy is given by the formula:
ΔKE = ½ * m * (v^2 - u^2)
Substituting the values, we get:
ΔKE = 0.5 * 1.63 kg * (15.0 m/s)^2
Now we need to find the work done against friction. The work done against friction is the force of friction multiplied by the distance over which the object slides.
To find the force of friction, we use the formula:
Frictional force = weight * sin(θ),
where θ is the angle the inclined plane makes with the horizontal.
The frictional force can be calculated as:
Frictional force = 16.0 N * sin(28.0°).
Next, we calculate the work done against friction:
Work = Frictional force * distance.
Plugging in the values, we get:
Work = (16.0 N * sin(28.0°)) * 150 m.
Now, using a calculator, we can determine the value for the work done against friction.